Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY CALCULUSf(x)QABCSFigure 2.2 A graph of a function, f(x), showing how differentiation correspondsto finding the gradient of the function at a particular point. Points B,Q and S are stationary points (see text).x◮Find the third derivative of the function f(x) =x 3 sin x.Using (2.14) we immediately findf ′′′ (x) =6sinx +3(6x)cosx +3(3x 2 )(− sin x)+x 3 (− cos x)=3(2− 3x 2 )sinx + x(18 − x 2 )cosx. ◭2.1.8 Special points of a functionWe have interpreted the derivative of a function as the gradient of the function atthe relevant point (figure 2.1). If the gradient is zero for some particular value ofx then the function is said to have a stationary point there. Clearly, in graphicalterms, this corresponds to a horizontal tangent to the graph.Stationary points may be divided into three categories and an example of eachis shown in figure 2.2. Point B is said to be a minimum since the function increasesin value in both directions away from it. Point Q is said to be a maximum sincethe function decreases in both directions away from it. Note that B is not theoverall minimum value of the function and Q is not the overall maximum; rather,they are a local minimum and a local maximum. Maxima and minima are knowncollectively as turning points.The third type of stationary point is the stationary point of inflection, S. Inthis case the function falls in the positive x-direction and rises in the negativex-direction so that S is neither a maximum nor a minimum. Nevertheless, thegradient of the function is zero at S, i.e. the graph of the function is flat there,and this justifies our calling it a stationary point. Of course, a point at which the50